483
Internat. J. Math. & Math. Sci.
VOLo 17 NO. 3 (1994) 483-488
SOME RESULTS ON INVARIANT APPROXIMATION
ANTONIO CARBONE
Dipartimento di Matematica
Universiti della Calabria
87036 Arcavacata di Rende (Cosenza)
Italy
(Received December 4, 1992 and in revised form May 10, 1993)
ABSTRACT. In this paper results on Invariant approximations, extending and unifying
earlier results are given. Several interesting results are derived as corollaries.
KEY WORDS AND PHRASES. Nonexpansive map, set of best approximants, starshaped
set, demiclosed, approximatively compact set.
1991 AMS MATHEMATICS SUBJECT CLASSIFICATION
CODES.
Primary 47H10,
Secondary 54H25.
1. INTRODUCTION
Several mathematicians have given results dealing with invariant approximation. The
earlier results due to Meinardus [7] and Brosowski [1] have been the main source of inspiration
for researchers in the field.
The following is due to Brosowski.
Let f be a contractive//near operator of a normed space X. Let Xo E X be an invariant
point and C an invariant set under f I[ the set of best C-approximants to xo is nonempty
compact and convex then it contains an f-invariaaat point.
Recently Carbone [2], [3], Habiniak [5], Hicks and Humphries [6], Narang [8], Po
and Mariadoss [9]
Sahab, Khan and Sessa [10], K.L.Singh [11], Singh [12], [13] and
Subrahmanyaan [15] gave extensions of the above theorem, and proved some further related
results.
The purpose of this paper is to extend and unify earlier results.
We need the following preliminaries.
If f: X
X where X is a normed linear space, and Ilfx fYl[
x Yll for x, y E X,
then f is called a nonexpansive map.
We denote by F(f), the set of fixed points of f. If x0 fxo, and Ilfx- f011 < I1 x011
for all x 5 X then f is said to be quasi-nonexpansive. A quasi-nonexpansive
map need not
be continuous.
Let C be a closed, convex subset of a normed linear space X. The set of best Capproximants to x consists of all y C such that
<-
Ilu 11 d(, C) inf{llz zll z e c}.
set C is said to be star-shaped if there exists at least one point p
C such that
A)p C for all x E C and 0 < A < 1. In this case p is called the star-center of C.
is said to be demiclosed if x,
x weakly and fx,
y strongly imply that y
fx.
The following result will be used in the proof of our theorem (see Subrahmanyam
[15],
Zhang [19]).
A. CARBONE
484
L(’t
Banach space" ad C o.mpt.r w(.,kly couq)act starshaped s,b,’t of X
C is onexp,si’’ and I- .f is demiclosed then f has a fixed point
X be
If f C
Note The
f(0C) C
a
all,we theorem h; been als
(,
f
pr(,ved when .f C
X is nonexpansive al eith,’,
is an inwar(l map.
n.elllark
X a nonexpansive ap witlt
In case C s a cnnpact, convex sutset of X and f C
(OC) C_ C, then f has a fixed point. This is valid even when C is star-shaped lint n,t
1.
n.cessarily cmvex.
-\
2 If C is xveakly compact convex and I-f is demiclosed then a nonexpansive map f C
with f(0C) C_ C has a fixed point.
We can replace convexity by star-shaped condition withut changing the conclusion.
normed linear space and M" is a subspace of X. and x0 E X, then d(.r0, M)
,,,/{llx0 yl] y e M} denotes the distance of x0 to the set M. PM(xo) {tl E M
II’ro 11 d(xo,M)} is the set of best approximations of .r0 in M. It is well km,wn that
the set P.u(x) is closed and convex for any subspace M of X (for details see [41).
If X is
a
MAIN RESULTS
Now we state our result.
X is such that f(OC) C_ (’
Let X be a Banach space. Suppose f" X
THEOREM
to Zo, is eat,’ly compact
of
best
set
X.
the
C-approximants
for
and f xo
I
D,
zo
Zo
2.
star-shaped and
i) f is nonexpansive
on
iii) I- f is demiclosed
then
f
has
a
D;
on
D
txed point closest to
PROOF First we show that
Let y G D. Then
f" D
3:o.
D, i.e. f is
a
self-map
on
D.
This implies that fy is also closest to x0 so fy co__ D.
Let p be the star center of D.
Define f,,.x k,,fx +(1-k,)p, where {k, } is a sequence of reals, 0 < k,, < 1 converging
cx for x
D.
to 1 as n
Each f, is contraction on D with contraction constant k, < 1, so ft:,.x,,.
x,,, for
n
1,2,. (see Subrahmanyam [15] for details).
Since D is weakly compact therefore {xk.} has a convergent subsequence convergint
:.
weakly to : say. For the sake of convenience let x,,
Now
Ilxt,
fx.,ll Ilft,x, Yx:,ll (k, 1)llfx,,ll + (1 k,)p 0 as
is bounded).
Since I- f is demiclosed so 0
(I- f):, i.e., 2 f’. Thus f has
fx,
a fixed point .r
closest to x0.
Note
If we take D as
a
compact convex set and
f D
D continuous then
we derive th,"
SOME RESULTS ON INVARIANT APPROXIMATION
same conclusion
f: DD. In
our
using Schauder fixed point theorem. In this case, we have to assume that
theorem f is nonexpansive so f: DD follows.
\Ve derive the fillwing rcsldts a. corollaries.
Corollary 1. If C is cnnpact, cmvex then I-
the esult (see
485
f dcmclsed is not
needed and
we
[19]).
Corollay 2. If C is weakly compact, convex and other conditions are the same tlwn
the result.
Recall that every convex set is starshaped. However, the converse is not true.
we get
In this section
we discuss a result of
Rao ad Mariadoss [9] given for approximatively
compact sets.
A set C C_ X is called approximatively compact if for each x G X and every sequence
in C with lim,,_oo IIz z,,
d(z, C), there exists a subsequence {x,, converging to
element of C.
A compact set is always approximatively compact but not conversely. A closed unit
{x,
an
ball in infinite dimensional Hilbert space is approximatively compact but not compact.
Let C be a nonempty approximatively compact subset of X and P" X 2 c a metric
projection of X onto C defined by Pc(x) {g C [[x g[I d(x, C)}.
Then Pc(x) 7 O, closed subset of C. In fact, Pc(x) is compact.
The following is due to Rao and Mariadoss [9].
Let X be a Banach space and f X X a continuous map.Let C be an approximatively
compact subset of X and an f-invariant set. Further, let f zo
zo, zo X and f satisfy
conditions:
the following
Ilfz fll < cl[ l[ / (11 fzll / II f[[) / (11 fy[[ / I1 fx[[)
for or,/3, 3’ positive with a + 2/3 + 27 < 1, for all z, g X.
If the set of best C-approximants to xo is nonempty starshaped then it has an
point closest to zo.
f-invariant
Note
x,
It is easy to derive that f satisfies the condition Ilfm- fm0[[ II m0ll for 1
(fXo x0), since c + 2/3 + 27 < 1, and since C is approximatively compact so the set of
=<
best C-approximants to z is compact.
Recently Carbone
THEOREM C
[3] proved the following:
Let f" X
X be a mapping on a Banach space X satisfying the following
fll < 11 x ull + (ll x fxl[ / IlY fYll)/ 7(11 x fYl[ + IlY fzll)
for a,/3, 7 positive with a, + 2/3 + 27 _< 1.
condition:
[Ifx
Let fxo z0, for x0 G X and C a subset of X such that f(OC) G C. If the set D of
best C-approximants to xo is dosed, strashaped and f(D) is compact with f continuous on
D, then f has a fixed point closest to xo.
We give the following to derive [91
as a
special case of
[31.
Remarks
1. / need not be continuous on the whole space X. Continuity only on D serves the purpose.
A. CARBONE
486
2. If C is approximatively compact then D is compact and f(D) is compact since a continuous inage of a compact set is compact.
3. C need not be f-invariant. Only f(c3C) C_ C is required.
In this sectin results in a locally convex Hausdorff topological vector space E are given.
L’t C be a nonempty subset of E and let p be a continuous seminorm on E. For .r E
define
d,,(z,C)
inf {p(:r
V) V
C}
and
Pc()
A mapping f C
{, e
c: p( u) a,(, c)}.
C is said to be p-nonexpansive if for all z,// C
p(fx-yy) <p(z-y),
(P: fanily of continuous seminorms); f C
such that
n
{k,} with k,
pe P
C is asymptotically nonexpansive if there
exists
p(f" z
f" y) k,p(z V)
for all z,
It is assumed that k,
d k,
C, p P, n 1,2,.
k,+ for
n
1,2,. [17].
f is uniformly asymptotically regular if for each p P and > 0 there exists a N(p, ) (=N
say) such that
p(f"z- f"+z) <
foall n > N and z C.
Vijayaraju
[18] proved the following:
THEOREM V
Let C be a nonempty subset of E.Let Zo E d f C C
ymptoticMly nonexpsive, uNformly ymptoticM1y rel map. Suppose that the set D of best
C-approximts to zo is nonempty, wely compact, stshaped d invt under f.
further, I- f is demidosed then f h a ed point closest to zo.
For further results on fixed points one should refer to [141, [161
We give the following in locally convex Hausdorff topological vector space E improving
Theorem V.
THEOREM 3
Let C be a subset of E, f xo
E
zo, :co E and f(OC) C_ C. Let f E
be an asymptoticM1y nonexpansive uniformly asymptotically regular map. Assume that the
set D of best C-approximants to xo is nonempty weakly compact, starshaped and invariant
under f. If I- f is demiclosed then f has a tixed point closest to :co.
Proof is on the same lines as in
Note
We do not need f" C
C
[18].
given in [18], only f(C)
Further related results are also given in [12].
Acknowledgement:
as is
_
thank Professor S.P Singh for hts encouragement and
C
serves
providing
the purpose.
the preprints/reprints
SOME RESULTS ON INVARIANT APPROXIMATION
487
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